# Properties

 Label 21.7.f.a Level $21$ Weight $7$ Character orbit 21.f Analytic conductor $4.831$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 21.f (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83113575602$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - x^{7} + 212 x^{6} + 473 x^{5} + 39800 x^{4} + 36821 x^{3} + 985651 x^{2} - 601290 x + 21068100$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3\cdot 7$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{1} + \beta_{2} ) q^{2} + ( -18 + 9 \beta_{2} ) q^{3} + ( 1 - 43 \beta_{2} + 4 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{4} + ( -26 + 2 \beta_{1} - 25 \beta_{2} - 5 \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{5} + ( 9 + 9 \beta_{1} - 18 \beta_{2} + 9 \beta_{3} ) q^{6} + ( -75 - 19 \beta_{1} - 12 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} - \beta_{5} + 5 \beta_{6} + 4 \beta_{7} ) q^{7} + ( 392 + 42 \beta_{1} + 4 \beta_{2} - 46 \beta_{3} + 9 \beta_{4} + 13 \beta_{5} + 12 \beta_{6} + 8 \beta_{7} ) q^{8} + ( 243 - 243 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{1} + \beta_{2} ) q^{2} + ( -18 + 9 \beta_{2} ) q^{3} + ( 1 - 43 \beta_{2} + 4 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{4} + ( -26 + 2 \beta_{1} - 25 \beta_{2} - 5 \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{5} + ( 9 + 9 \beta_{1} - 18 \beta_{2} + 9 \beta_{3} ) q^{6} + ( -75 - 19 \beta_{1} - 12 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} - \beta_{5} + 5 \beta_{6} + 4 \beta_{7} ) q^{7} + ( 392 + 42 \beta_{1} + 4 \beta_{2} - 46 \beta_{3} + 9 \beta_{4} + 13 \beta_{5} + 12 \beta_{6} + 8 \beta_{7} ) q^{8} + ( 243 - 243 \beta_{2} ) q^{9} + ( -593 + 60 \beta_{1} + 299 \beta_{2} - 32 \beta_{3} - 14 \beta_{4} - \beta_{5} - \beta_{6} + 9 \beta_{7} ) q^{10} + ( -12 - \beta_{1} - 44 \beta_{2} + 105 \beta_{3} + 3 \beta_{4} + 11 \beta_{5} - 9 \beta_{6} + 13 \beta_{7} ) q^{11} + ( 369 + 36 \beta_{1} + 387 \beta_{2} - 72 \beta_{3} + 9 \beta_{4} + 18 \beta_{5} - 9 \beta_{6} + 9 \beta_{7} ) q^{12} + ( 303 + 3 \beta_{1} - 567 \beta_{2} + 2 \beta_{3} + 18 \beta_{4} - 19 \beta_{5} + 39 \beta_{6} - 40 \beta_{7} ) q^{13} + ( 249 + 122 \beta_{1} - 1844 \beta_{2} + 31 \beta_{3} - 25 \beta_{4} - 30 \beta_{5} + 25 \beta_{6} - 13 \beta_{7} ) q^{14} + ( 702 - 63 \beta_{1} - 9 \beta_{2} + 72 \beta_{3} - 9 \beta_{5} - 27 \beta_{6} - 18 \beta_{7} ) q^{15} + ( -3015 - 626 \beta_{1} + 3035 \beta_{2} + 20 \beta_{3} + 5 \beta_{4} - 40 \beta_{5} - 55 \beta_{6} + 35 \beta_{7} ) q^{16} + ( -896 - 182 \beta_{1} + 436 \beta_{2} + 82 \beta_{3} + 30 \beta_{4} + 42 \beta_{5} + 42 \beta_{6} - 6 \beta_{7} ) q^{17} + ( 243 \beta_{2} - 243 \beta_{3} ) q^{18} + ( -1425 + 166 \beta_{1} - 1594 \beta_{2} - 251 \beta_{3} - 125 \beta_{4} - 88 \beta_{5} - 118 \beta_{6} - 125 \beta_{7} ) q^{19} + ( -1482 + 366 \beta_{1} + 2922 \beta_{2} + 354 \beta_{3} - 39 \beta_{4} + 27 \beta_{5} - 42 \beta_{6} + 30 \beta_{7} ) q^{20} + ( 1503 + 225 \beta_{1} - 576 \beta_{2} + 63 \beta_{3} - 63 \beta_{4} - 27 \beta_{5} - 99 \beta_{6} - 81 \beta_{7} ) q^{21} + ( 10930 - 220 \beta_{1} - 44 \beta_{2} + 264 \beta_{3} + 37 \beta_{4} - 7 \beta_{5} - 132 \beta_{6} - 88 \beta_{7} ) q^{22} + ( 944 - 344 \beta_{1} - 1008 \beta_{2} - 64 \beta_{3} - 144 \beta_{4} + 128 \beta_{5} + 48 \beta_{6} + 16 \beta_{7} ) q^{23} + ( -7137 - 792 \beta_{1} + 3609 \beta_{2} + 450 \beta_{3} - 54 \beta_{4} - 189 \beta_{5} - 189 \beta_{6} - 27 \beta_{7} ) q^{24} + ( -60 + 43 \beta_{1} - 3927 \beta_{2} + 1021 \beta_{3} - 129 \beta_{4} + 103 \beta_{5} - 189 \beta_{6} + 17 \beta_{7} ) q^{25} + ( 358 + 1221 \beta_{1} + 804 \beta_{2} - 2678 \beta_{3} + 341 \beta_{4} + 210 \beta_{5} + 367 \beta_{6} + 341 \beta_{7} ) q^{26} + ( -2187 + 4374 \beta_{2} ) q^{27} + ( 5681 + 2932 \beta_{1} + 8285 \beta_{2} - 2154 \beta_{3} + 261 \beta_{4} + 194 \beta_{5} + 71 \beta_{6} + 225 \beta_{7} ) q^{28} + ( -1040 - 1537 \beta_{1} + 145 \beta_{2} + 1392 \beta_{3} - 48 \beta_{4} + 97 \beta_{5} + 435 \beta_{6} + 290 \beta_{7} ) q^{29} + ( 8073 - 828 \beta_{1} - 8037 \beta_{2} + 36 \beta_{3} + 243 \beta_{4} - 72 \beta_{5} + 135 \beta_{6} - 171 \beta_{7} ) q^{30} + ( -14670 - 1086 \beta_{1} + 7313 \beta_{2} + 530 \beta_{3} + 62 \beta_{4} + 70 \beta_{5} + 70 \beta_{6} - 18 \beta_{7} ) q^{31} + ( 465 - 116 \beta_{1} - 45905 \beta_{2} + 4330 \beta_{3} + 348 \beta_{4} - 581 \beta_{5} + 813 \beta_{6} - 349 \beta_{7} ) q^{32} + ( 630 + 954 \beta_{1} + 387 \beta_{2} - 1881 \beta_{3} - 135 \beta_{4} - 216 \beta_{5} + 54 \beta_{6} - 135 \beta_{7} ) q^{33} + ( 8976 - 464 \beta_{1} - 17884 \beta_{2} - 248 \beta_{3} + 358 \beta_{4} - 142 \beta_{5} + 68 \beta_{6} + 148 \beta_{7} ) q^{34} + ( -19652 + 1261 \beta_{1} + 32617 \beta_{2} + 804 \beta_{3} - 344 \beta_{4} + 47 \beta_{5} - 127 \beta_{6} + 50 \beta_{7} ) q^{35} + ( -10206 - 972 \beta_{1} + 972 \beta_{3} - 243 \beta_{4} - 243 \beta_{5} ) q^{36} + ( 16431 - 1368 \beta_{1} - 16544 \beta_{2} - 113 \beta_{3} - 79 \beta_{4} + 226 \beta_{5} + 260 \beta_{6} - 147 \beta_{7} ) q^{37} + ( -14333 + 5074 \beta_{1} + 7018 \beta_{2} - 2639 \beta_{3} + 390 \beta_{4} + 501 \beta_{5} + 501 \beta_{6} - 93 \beta_{7} ) q^{38} + ( -540 - 9 \beta_{1} + 7659 \beta_{2} - 63 \beta_{3} + 27 \beta_{4} + 531 \beta_{5} - 513 \beta_{6} + 549 \beta_{7} ) q^{39} + ( 18317 + 586 \beta_{1} + 18335 \beta_{2} - 1200 \beta_{3} + 23 \beta_{4} - 10 \beta_{5} + 61 \beta_{6} + 23 \beta_{7} ) q^{40} + ( -8866 - 1052 \beta_{1} + 16648 \beta_{2} - 1272 \beta_{3} - 872 \beta_{4} + 652 \beta_{5} - 1084 \beta_{6} + 864 \beta_{7} ) q^{41} + ( 12267 - 819 \beta_{1} + 18567 \beta_{2} - 1656 \beta_{3} + 675 \beta_{4} + 387 \beta_{5} - 36 \beta_{7} ) q^{42} + ( 6845 - 1459 \beta_{1} - 221 \beta_{2} + 1680 \beta_{3} - 620 \beta_{4} - 841 \beta_{5} - 663 \beta_{6} - 442 \beta_{7} ) q^{43} + ( 25623 - 7226 \beta_{1} - 25235 \beta_{2} + 388 \beta_{3} + 783 \beta_{4} - 776 \beta_{5} - 381 \beta_{6} - 7 \beta_{7} ) q^{44} + ( -12636 + 1215 \beta_{1} + 6318 \beta_{2} - 729 \beta_{3} - 243 \beta_{4} + 243 \beta_{5} + 243 \beta_{6} + 243 \beta_{7} ) q^{45} + ( -344 + 64 \beta_{1} - 27616 \beta_{2} - 184 \beta_{3} - 192 \beta_{4} + 408 \beta_{5} - 536 \beta_{6} + 280 \beta_{7} ) q^{46} + ( 41226 + 1220 \beta_{1} + 39896 \beta_{2} - 1302 \beta_{3} - 1234 \beta_{4} - 192 \beta_{5} - 2180 \beta_{6} - 1234 \beta_{7} ) q^{47} + ( 27630 + 5814 \beta_{1} - 54810 \beta_{2} + 5274 \beta_{3} - 585 \beta_{4} + 45 \beta_{5} + 450 \beta_{6} - 990 \beta_{7} ) q^{48} + ( -66640 + 5525 \beta_{1} + 55496 \beta_{2} + 1278 \beta_{3} - 178 \beta_{4} - 637 \beta_{5} + 985 \beta_{6} - 1008 \beta_{7} ) q^{49} + ( 121141 + 2511 \beta_{1} - 412 \beta_{2} - 2099 \beta_{3} + 1065 \beta_{4} + 653 \beta_{5} - 1236 \beta_{6} - 824 \beta_{7} ) q^{50} + ( 11772 + 2376 \beta_{1} - 11610 \beta_{2} + 162 \beta_{3} - 162 \beta_{4} - 324 \beta_{5} - 648 \beta_{6} + 486 \beta_{7} ) q^{51} + ( -275506 - 14552 \beta_{1} + 138340 \beta_{2} + 7402 \beta_{3} - 2096 \beta_{4} - 1426 \beta_{5} - 1426 \beta_{6} + 922 \beta_{7} ) q^{52} + ( 864 + 511 \beta_{1} - 137478 \beta_{2} - 7879 \beta_{3} - 1533 \beta_{4} - 353 \beta_{5} - 669 \beta_{6} - 1375 \beta_{7} ) q^{53} + ( -2187 - 2187 \beta_{1} - 2187 \beta_{2} + 4374 \beta_{3} ) q^{54} + ( 46156 - 7433 \beta_{1} - 90141 \beta_{2} - 7102 \beta_{3} + 1582 \beta_{4} - 1251 \beta_{5} + 2171 \beta_{6} - 1840 \beta_{7} ) q^{55} + ( -137333 - 20804 \beta_{1} + 221837 \beta_{2} + 3878 \beta_{3} - 1316 \beta_{4} - 2359 \beta_{5} - 1393 \beta_{6} + 665 \beta_{7} ) q^{56} + ( 39663 - 3753 \beta_{1} + 729 \beta_{2} + 3024 \beta_{3} + 1188 \beta_{4} + 1917 \beta_{5} + 2187 \beta_{6} + 1458 \beta_{7} ) q^{57} + ( 130129 + 13414 \beta_{1} - 130901 \beta_{2} - 772 \beta_{3} - 1699 \beta_{4} + 1544 \beta_{5} + 617 \beta_{6} + 155 \beta_{7} ) q^{58} + ( -27852 + 2709 \beta_{1} + 13548 \beta_{2} - 571 \beta_{3} + 3079 \beta_{4} - 811 \beta_{5} - 811 \beta_{6} - 2323 \beta_{7} ) q^{59} + ( 405 - 108 \beta_{1} - 39393 \beta_{2} - 9666 \beta_{3} + 324 \beta_{4} - 513 \beta_{5} + 729 \beta_{6} - 297 \beta_{7} ) q^{60} + ( 36236 - 17232 \beta_{1} + 39700 \beta_{2} + 32440 \beta_{3} + 2744 \beta_{4} + 1440 \beta_{5} + 3328 \beta_{6} + 2744 \beta_{7} ) q^{61} + ( 63657 + 6697 \beta_{1} - 126374 \beta_{2} + 7121 \beta_{3} + 1106 \beta_{4} - 682 \beta_{5} + 940 \beta_{6} - 516 \beta_{7} ) q^{62} + ( -22356 - 1458 \beta_{1} + 18468 \beta_{2} - 3159 \beta_{3} + 243 \beta_{4} + 972 \beta_{5} + 1458 \beta_{6} + 1215 \beta_{7} ) q^{63} + ( 281992 + 47654 \beta_{1} + 1044 \beta_{2} - 48698 \beta_{3} + 2207 \beta_{4} + 3251 \beta_{5} + 3132 \beta_{6} + 2088 \beta_{7} ) q^{64} + ( -111258 + 24248 \beta_{1} + 111212 \beta_{2} - 46 \beta_{3} - 2178 \beta_{4} + 92 \beta_{5} - 2040 \beta_{6} + 2086 \beta_{7} ) q^{65} + ( -197073 + 4356 \beta_{1} + 98703 \beta_{2} - 2772 \beta_{3} - 1854 \beta_{4} + 855 \beta_{5} + 855 \beta_{6} + 1521 \beta_{7} ) q^{66} + ( -2148 - 1045 \beta_{1} - 289965 \beta_{2} - 4267 \beta_{3} + 3135 \beta_{4} + 1103 \beta_{5} + 987 \beta_{6} + 3193 \beta_{7} ) q^{67} + ( -69942 + 156 \beta_{1} - 68274 \beta_{2} - 1440 \beta_{3} + 1398 \beta_{4} + 540 \beta_{5} + 1986 \beta_{6} + 1398 \beta_{7} ) q^{68} + ( -8928 + 2520 \beta_{1} + 18720 \beta_{2} + 4248 \beta_{3} + 3024 \beta_{4} - 1296 \beta_{5} + 864 \beta_{6} + 864 \beta_{7} ) q^{69} + ( 81733 - 6836 \beta_{1} + 115725 \beta_{2} + 30590 \beta_{3} - 413 \beta_{4} + 3300 \beta_{5} - 2341 \beta_{6} + 737 \beta_{7} ) q^{70} + ( 188446 - 20866 \beta_{1} - 758 \beta_{2} + 21624 \beta_{3} - 3408 \beta_{4} - 4166 \beta_{5} - 2274 \beta_{6} - 1516 \beta_{7} ) q^{71} + ( 97443 + 11178 \beta_{1} - 98415 \beta_{2} - 972 \beta_{3} - 729 \beta_{4} + 1944 \beta_{5} + 2187 \beta_{6} - 1215 \beta_{7} ) q^{72} + ( -71634 + 27083 \beta_{1} + 34855 \beta_{2} - 14317 \beta_{3} + 2297 \beta_{4} + 3475 \beta_{5} + 3475 \beta_{6} - 373 \beta_{7} ) q^{73} + ( -723 - 588 \beta_{1} - 113570 \beta_{2} - 21745 \beta_{3} + 1764 \beta_{4} + 135 \beta_{5} + 1041 \beta_{6} + 1311 \beta_{7} ) q^{74} + ( 35649 + 8802 \beta_{1} + 35730 \beta_{2} - 18765 \beta_{3} + 621 \beta_{4} - 1080 \beta_{5} + 2862 \beta_{6} + 621 \beta_{7} ) q^{75} + ( -178434 - 20802 \beta_{1} + 351000 \beta_{2} - 23238 \beta_{3} - 6588 \beta_{4} + 4152 \beta_{5} - 5868 \beta_{6} + 3432 \beta_{7} ) q^{76} + ( 131974 - 22686 \beta_{1} + 9761 \beta_{2} - 5245 \beta_{3} + 2777 \beta_{4} + 1416 \beta_{5} + 3562 \beta_{6} - 4999 \beta_{7} ) q^{77} + ( -12501 - 35091 \beta_{1} - 2124 \beta_{2} + 37215 \beta_{3} - 2835 \beta_{4} - 4959 \beta_{5} - 6372 \beta_{6} - 4248 \beta_{7} ) q^{78} + ( 160501 - 5248 \beta_{1} - 157751 \beta_{2} + 2750 \beta_{3} + 7518 \beta_{4} - 5500 \beta_{5} - 732 \beta_{6} - 2018 \beta_{7} ) q^{79} + ( -353365 + 7040 \beta_{1} + 177485 \beta_{2} - 3990 \beta_{3} - 4150 \beta_{4} - 665 \beta_{5} - 665 \beta_{6} + 2545 \beta_{7} ) q^{80} -59049 \beta_{2} q^{81} + ( -111998 - 34314 \beta_{1} - 123606 \beta_{2} + 74692 \beta_{3} - 8836 \beta_{4} - 5544 \beta_{5} - 9356 \beta_{6} - 8836 \beta_{7} ) q^{82} + ( -97070 - 15973 \beta_{1} + 192851 \beta_{2} - 15078 \beta_{3} + 698 \beta_{4} + 197 \beta_{5} - 1289 \beta_{6} + 2184 \beta_{7} ) q^{83} + ( -176544 - 45774 \beta_{1} - 21690 \beta_{2} + 12384 \beta_{3} - 4059 \beta_{4} - 3771 \beta_{5} - 2988 \beta_{6} - 2304 \beta_{7} ) q^{84} + ( -38232 + 22438 \beta_{1} - 326 \beta_{2} - 22112 \beta_{3} - 3480 \beta_{4} - 3806 \beta_{5} - 978 \beta_{6} - 652 \beta_{7} ) q^{85} + ( 198682 + 24517 \beta_{1} - 200278 \beta_{2} - 1596 \beta_{3} + 1539 \beta_{4} + 3192 \beta_{5} + 6327 \beta_{6} - 4731 \beta_{7} ) q^{86} + ( 19152 + 26361 \beta_{1} - 9792 \beta_{2} - 11223 \beta_{3} + 4779 \beta_{4} - 3483 \beta_{5} - 3483 \beta_{6} - 4347 \beta_{7} ) q^{87} + ( 9479 + 2788 \beta_{1} - 92535 \beta_{2} + 16374 \beta_{3} - 8364 \beta_{4} - 6691 \beta_{5} + 1115 \beta_{6} - 12267 \beta_{7} ) q^{88} + ( -81340 - 2428 \beta_{1} - 81682 \beta_{2} + 5582 \beta_{3} - 534 \beta_{4} + 384 \beta_{5} - 1644 \beta_{6} - 534 \beta_{7} ) q^{89} + ( -73872 + 7776 \beta_{1} + 144342 \beta_{2} + 6804 \beta_{3} - 3159 \beta_{4} + 2187 \beta_{5} - 3402 \beta_{6} + 2430 \beta_{7} ) q^{90} + ( -76547 + 76118 \beta_{1} + 193604 \beta_{2} - 70667 \beta_{3} + 2651 \beta_{4} - 4416 \beta_{5} - 5894 \beta_{6} + 5827 \beta_{7} ) q^{91} + ( 82320 - 18704 \beta_{1} + 2464 \beta_{2} + 16240 \beta_{3} + 1680 \beta_{4} + 4144 \beta_{5} + 7392 \beta_{6} + 4928 \beta_{7} ) q^{92} + ( 197451 + 14544 \beta_{1} - 197217 \beta_{2} + 234 \beta_{3} - 486 \beta_{4} - 468 \beta_{5} - 1188 \beta_{6} + 954 \beta_{7} ) q^{93} + ( -86578 - 101972 \beta_{1} + 41952 \beta_{2} + 52686 \beta_{3} + 8748 \beta_{4} - 726 \beta_{5} - 726 \beta_{6} - 6074 \beta_{7} ) q^{94} + ( -4680 + 1292 \beta_{1} - 137338 \beta_{2} + 24524 \beta_{3} - 3876 \beta_{4} + 5972 \beta_{5} - 8556 \beta_{6} + 3388 \beta_{7} ) q^{95} + ( 406863 + 40014 \beta_{1} + 412101 \beta_{2} - 76896 \beta_{3} + 1053 \beta_{4} + 8370 \beta_{5} - 10449 \beta_{6} + 1053 \beta_{7} ) q^{96} + ( -118308 - 33555 \beta_{1} + 232593 \beta_{2} - 37586 \beta_{3} - 8058 \beta_{4} + 4027 \beta_{5} - 4023 \beta_{6} - 8 \beta_{7} ) q^{97} + ( 150010 + 58739 \beta_{1} + 531307 \beta_{2} - 18889 \beta_{3} + 7291 \beta_{4} + 16086 \beta_{5} + 4121 \beta_{6} + 14371 \beta_{7} ) q^{98} + ( -14094 - 25515 \beta_{1} + 243 \beta_{2} + 25272 \beta_{3} + 2916 \beta_{4} + 3159 \beta_{5} + 729 \beta_{6} + 486 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 5 q^{2} - 108 q^{3} - 173 q^{4} - 294 q^{5} - 656 q^{7} + 3326 q^{8} + 972 q^{9} + O(q^{10})$$ $$8 q - 5 q^{2} - 108 q^{3} - 173 q^{4} - 294 q^{5} - 656 q^{7} + 3326 q^{8} + 972 q^{9} - 3411 q^{10} - 314 q^{11} + 4671 q^{12} - 5360 q^{14} + 5292 q^{15} - 12721 q^{16} - 5532 q^{17} + 1215 q^{18} - 18234 q^{19} + 9342 q^{21} + 86106 q^{22} + 3928 q^{23} - 44901 q^{24} - 17038 q^{25} + 12366 q^{26} + 85037 q^{28} - 8300 q^{29} + 30699 q^{30} - 89508 q^{31} - 186207 q^{32} + 8478 q^{33} - 25860 q^{35} - 84078 q^{36} + 64706 q^{37} - 77136 q^{38} + 29106 q^{39} + 221823 q^{40} + 173583 q^{42} + 45740 q^{43} + 92529 q^{44} - 71442 q^{45} - 111504 q^{46} + 483276 q^{47} - 310684 q^{49} + 967216 q^{50} + 49788 q^{51} - 1673988 q^{52} - 540974 q^{53} - 32805 q^{54} - 241885 q^{56} + 328212 q^{57} + 539799 q^{58} - 181770 q^{59} - 146367 q^{60} + 418224 q^{61} - 92826 q^{63} + 2378626 q^{64} - 414204 q^{65} - 1162431 q^{66} - 1158902 q^{67} - 821250 q^{68} + 1087917 q^{70} + 1442344 q^{71} + 404109 q^{72} - 378666 q^{73} - 432940 q^{74} + 460026 q^{75} + 1065994 q^{77} - 222588 q^{78} + 611452 q^{79} - 2094945 q^{80} - 236196 q^{81} - 1561266 q^{82} - 1579716 q^{84} - 275112 q^{85} + 816224 q^{86} + 112050 q^{87} - 366441 q^{88} - 989196 q^{89} + 304446 q^{91} + 678720 q^{92} + 805572 q^{93} - 716148 q^{94} - 591792 q^{95} + 5027589 q^{96} + 3509629 q^{98} - 152604 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 212 x^{6} + 473 x^{5} + 39800 x^{4} + 36821 x^{3} + 985651 x^{2} - 601290 x + 21068100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-405518177 \nu^{7} - 12595814413 \nu^{6} - 65484675874 \nu^{5} - 2673339327091 \nu^{4} - 20860832470090 \nu^{3} - 529666993003657 \nu^{2} - 423382833624317 \nu - 1017283529220570$$$$)/ 11540026222043130$$ $$\beta_{3}$$ $$=$$ $$($$$$8497603 \nu^{7} - 13389005 \nu^{6} + 1621914529 \nu^{5} + 3085757533 \nu^{4} + 336428371378 \nu^{3} + 15479370553 \nu^{2} + 8366760637851 \nu - 5583985297290$$$$)/ 7542500798721$$ $$\beta_{4}$$ $$=$$ $$($$$$30359760961 \nu^{7} - 17272473736 \nu^{6} + 12542253415247 \nu^{5} - 45647117796292 \nu^{4} + 2374626244486355 \nu^{3} - 4649039539643404 \nu^{2} + 177824901485458486 \nu - 887386378854226320$$$$)/ 23080052444086260$$ $$\beta_{5}$$ $$=$$ $$($$$$36613259519 \nu^{7} + 484838322556 \nu^{6} + 240706081393 \nu^{5} + 69967211645572 \nu^{4} + 594392649047605 \nu^{3} + 4771038672935884 \nu^{2} - 171328578806357686 \nu - 1022366367515487780$$$$)/ 23080052444086260$$ $$\beta_{6}$$ $$=$$ $$($$$$20960695316 \nu^{7} - 201659710511 \nu^{6} + 3951590320162 \nu^{5} - 21226630078907 \nu^{4} + 569010508767670 \nu^{3} - 4885911718111139 \nu^{2} - 9572214805242754 \nu - 12764692598290560$$$$)/ 3297150349155180$$ $$\beta_{7}$$ $$=$$ $$($$$$14593340681 \nu^{7} + 40700920519 \nu^{6} + 3013487903677 \nu^{5} + 21592891249843 \nu^{4} + 580595236989325 \nu^{3} + 2960686065927031 \nu^{2} + 14428518294215396 \nu + 66916687643934480$$$$)/ 1357650143769780$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{3} - 106 \beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$-8 \beta_{7} - 12 \beta_{6} - 10 \beta_{5} - 6 \beta_{4} + 165 \beta_{3} - 4 \beta_{2} - 161 \beta_{1} - 204$$ $$\nu^{4}$$ $$=$$ $$213 \beta_{7} - 217 \beta_{6} - 8 \beta_{5} - 205 \beta_{4} + 4 \beta_{3} + 17990 \beta_{2} - 718 \beta_{1} - 17986$$ $$\nu^{5}$$ $$=$$ $$2462 \beta_{7} + 930 \beta_{6} + 766 \beta_{5} + 2544 \beta_{4} - 30357 \beta_{3} + 77878 \beta_{2} - 848 \beta_{1} - 1614$$ $$\nu^{6}$$ $$=$$ $$4432 \beta_{7} + 6648 \beta_{6} + 45813 \beta_{5} + 43597 \beta_{4} - 195138 \beta_{3} + 2216 \beta_{2} + 192922 \beta_{1} + 3368601$$ $$\nu^{7}$$ $$=$$ $$-221730 \beta_{7} + 385038 \beta_{6} + 326616 \beta_{5} - 104886 \beta_{4} - 163308 \beta_{3} - 21042174 \beta_{2} + 6116101 \beta_{1} + 20878866$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/21\mathbb{Z}\right)^\times$$.

 $$n$$ $$8$$ $$10$$ $$\chi(n)$$ $$1$$ $$1 - \beta_{2}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 7.29767 + 12.6399i 2.26350 + 3.92050i −2.75320 − 4.76869i −6.30797 − 10.9257i 7.29767 − 12.6399i 2.26350 − 3.92050i −2.75320 + 4.76869i −6.30797 + 10.9257i
−7.79767 13.5060i −13.5000 7.79423i −89.6073 + 155.204i 22.3721 12.9165i 243.107i −203.121 276.389i 1796.81 121.500 + 210.444i −348.901 201.438i
10.2 −2.76350 4.78652i −13.5000 7.79423i 16.7261 28.9705i −57.9943 + 33.4830i 86.1574i −240.457 + 244.600i −538.619 121.500 + 210.444i 320.535 + 185.061i
10.3 2.25320 + 3.90266i −13.5000 7.79423i 21.8461 37.8386i 53.9244 31.1333i 70.2479i 218.833 264.124i 485.305 121.500 + 210.444i 243.005 + 140.299i
10.4 5.80797 + 10.0597i −13.5000 7.79423i −35.4650 + 61.4271i −165.302 + 95.4373i 181.074i −103.254 + 327.089i −80.4975 121.500 + 210.444i −1920.14 1108.59i
19.1 −7.79767 + 13.5060i −13.5000 + 7.79423i −89.6073 155.204i 22.3721 + 12.9165i 243.107i −203.121 + 276.389i 1796.81 121.500 210.444i −348.901 + 201.438i
19.2 −2.76350 + 4.78652i −13.5000 + 7.79423i 16.7261 + 28.9705i −57.9943 33.4830i 86.1574i −240.457 244.600i −538.619 121.500 210.444i 320.535 185.061i
19.3 2.25320 3.90266i −13.5000 + 7.79423i 21.8461 + 37.8386i 53.9244 + 31.1333i 70.2479i 218.833 + 264.124i 485.305 121.500 210.444i 243.005 140.299i
19.4 5.80797 10.0597i −13.5000 + 7.79423i −35.4650 61.4271i −165.302 95.4373i 181.074i −103.254 327.089i −80.4975 121.500 210.444i −1920.14 + 1108.59i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 19.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.7.f.a 8
3.b odd 2 1 63.7.m.d 8
4.b odd 2 1 336.7.bh.d 8
7.b odd 2 1 147.7.f.d 8
7.c even 3 1 147.7.d.b 8
7.c even 3 1 147.7.f.d 8
7.d odd 6 1 inner 21.7.f.a 8
7.d odd 6 1 147.7.d.b 8
21.g even 6 1 63.7.m.d 8
21.g even 6 1 441.7.d.c 8
21.h odd 6 1 441.7.d.c 8
28.f even 6 1 336.7.bh.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.f.a 8 1.a even 1 1 trivial
21.7.f.a 8 7.d odd 6 1 inner
63.7.m.d 8 3.b odd 2 1
63.7.m.d 8 21.g even 6 1
147.7.d.b 8 7.c even 3 1
147.7.d.b 8 7.d odd 6 1
147.7.f.d 8 7.b odd 2 1
147.7.f.d 8 7.c even 3 1
336.7.bh.d 8 4.b odd 2 1
336.7.bh.d 8 28.f even 6 1
441.7.d.c 8 21.g even 6 1
441.7.d.c 8 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + \cdots$$ acting on $$S_{7}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$20358144 - 1281408 T + 992080 T^{2} + 12248 T^{3} + 37712 T^{4} - 442 T^{5} + 227 T^{6} + 5 T^{7} + T^{8}$$
$3$ $$( 243 + 27 T + T^{2} )^{4}$$
$5$ $$422734160250000 - 25332592050000 T + 334857307500 T^{2} + 10257232500 T^{3} - 72000675 T^{4} - 2447550 T^{5} + 20487 T^{6} + 294 T^{7} + T^{8}$$
$7$ $$19\!\cdots\!01$$$$+ 1068239320229254544 T + 5128335320842510 T^{2} + 12493153898344 T^{3} + 41840034887 T^{4} + 106190056 T^{5} + 370510 T^{6} + 656 T^{7} + T^{8}$$
$11$ $$18\!\cdots\!44$$$$+ 98293722790383421872 T + 5063514049337341372 T^{2} + 8590189886857868 T^{3} + 14795841919805 T^{4} + 3395131754 T^{5} + 3844943 T^{6} + 314 T^{7} + T^{8}$$
$13$ $$21\!\cdots\!56$$$$+$$$$14\!\cdots\!08$$$$T^{2} + 331924469476329 T^{4} + 30553062 T^{6} + T^{8}$$
$17$ $$14\!\cdots\!04$$$$+$$$$32\!\cdots\!48$$$$T - 18954316226540153088 T^{2} - 49044433474028160 T^{3} + 325284835782096 T^{4} - 99181347120 T^{5} - 7727652 T^{6} + 5532 T^{7} + T^{8}$$
$19$ $$40\!\cdots\!56$$$$+$$$$29\!\cdots\!56$$$$T +$$$$80\!\cdots\!44$$$$T^{2} + 80019415606831775892 T^{3} - 3659130219753963 T^{4} - 1012664156058 T^{5} + 55289115 T^{6} + 18234 T^{7} + T^{8}$$
$23$ $$15\!\cdots\!56$$$$-$$$$35\!\cdots\!28$$$$T +$$$$83\!\cdots\!96$$$$T^{2} + 12921467685980241920 T^{3} + 20933343724396544 T^{4} + 749772681728 T^{5} + 161320832 T^{6} - 3928 T^{7} + T^{8}$$
$29$ $$( -131918946476762880 - 26271861557728 T - 1346898667 T^{2} + 4150 T^{3} + T^{4} )^{2}$$
$31$ $$16\!\cdots\!01$$$$-$$$$45\!\cdots\!96$$$$T -$$$$58\!\cdots\!94$$$$T^{2} +$$$$26\!\cdots\!36$$$$T^{3} + 706793352941013399 T^{4} + 68718078342072 T^{5} + 3438291822 T^{6} + 89508 T^{7} + T^{8}$$
$37$ $$23\!\cdots\!36$$$$-$$$$16\!\cdots\!60$$$$T +$$$$13\!\cdots\!00$$$$T^{2} -$$$$47\!\cdots\!28$$$$T^{3} + 2903640625281916109 T^{4} - 97788930340930 T^{5} + 3733577311 T^{6} - 64706 T^{7} + T^{8}$$
$41$ $$36\!\cdots\!44$$$$+$$$$43\!\cdots\!00$$$$T^{2} +$$$$15\!\cdots\!32$$$$T^{4} + 21593029296 T^{6} + T^{8}$$
$43$ $$( -293597400770976796 + 85125133057276 T - 4841210811 T^{2} - 22870 T^{3} + T^{4} )^{2}$$
$47$ $$99\!\cdots\!96$$$$-$$$$26\!\cdots\!80$$$$T +$$$$10\!\cdots\!60$$$$T^{2} +$$$$36\!\cdots\!00$$$$T^{3} -$$$$15\!\cdots\!24$$$$T^{4} - 6508813429466640 T^{5} + 91320005532 T^{6} - 483276 T^{7} + T^{8}$$
$53$ $$73\!\cdots\!64$$$$-$$$$30\!\cdots\!60$$$$T +$$$$20\!\cdots\!00$$$$T^{2} +$$$$41\!\cdots\!16$$$$T^{3} +$$$$60\!\cdots\!97$$$$T^{4} + 40803243733591190 T^{5} + 203978340551 T^{6} + 540974 T^{7} + T^{8}$$
$59$ $$20\!\cdots\!00$$$$+$$$$67\!\cdots\!60$$$$T -$$$$32\!\cdots\!28$$$$T^{2} -$$$$11\!\cdots\!68$$$$T^{3} +$$$$51\!\cdots\!49$$$$T^{4} - 13446651706756290 T^{5} - 62962743777 T^{6} + 181770 T^{7} + T^{8}$$
$61$ $$68\!\cdots\!00$$$$+$$$$30\!\cdots\!00$$$$T -$$$$93\!\cdots\!00$$$$T^{2} -$$$$63\!\cdots\!00$$$$T^{3} +$$$$15\!\cdots\!00$$$$T^{4} + 71069068838315520 T^{5} - 111626861088 T^{6} - 418224 T^{7} + T^{8}$$
$67$ $$29\!\cdots\!04$$$$+$$$$10\!\cdots\!28$$$$T +$$$$26\!\cdots\!52$$$$T^{2} +$$$$21\!\cdots\!00$$$$T^{3} +$$$$12\!\cdots\!29$$$$T^{4} + 387875645278331366 T^{5} + 908045258107 T^{6} + 1158902 T^{7} + T^{8}$$
$71$ $$( -$$$$19\!\cdots\!12$$$$+ 115286062429846880 T - 36558200716 T^{2} - 721172 T^{3} + T^{4} )^{2}$$
$73$ $$77\!\cdots\!24$$$$+$$$$13\!\cdots\!92$$$$T +$$$$10\!\cdots\!44$$$$T^{2} +$$$$36\!\cdots\!56$$$$T^{3} +$$$$43\!\cdots\!01$$$$T^{4} - 88270688635072434 T^{5} - 185313643497 T^{6} + 378666 T^{7} + T^{8}$$
$79$ $$99\!\cdots\!01$$$$-$$$$96\!\cdots\!32$$$$T +$$$$92\!\cdots\!06$$$$T^{2} -$$$$15\!\cdots\!80$$$$T^{3} +$$$$81\!\cdots\!39$$$$T^{4} - 103324245868979728 T^{5} + 521211945622 T^{6} - 611452 T^{7} + T^{8}$$
$83$ $$52\!\cdots\!24$$$$+$$$$31\!\cdots\!16$$$$T^{2} +$$$$85\!\cdots\!29$$$$T^{4} + 524747194014 T^{6} + T^{8}$$
$89$ $$87\!\cdots\!56$$$$-$$$$92\!\cdots\!00$$$$T +$$$$22\!\cdots\!76$$$$T^{2} +$$$$10\!\cdots\!00$$$$T^{3} +$$$$14\!\cdots\!80$$$$T^{4} + 103856130641901456 T^{5} + 431160022908 T^{6} + 989196 T^{7} + T^{8}$$
$97$ $$90\!\cdots\!76$$$$+$$$$26\!\cdots\!08$$$$T^{2} +$$$$12\!\cdots\!73$$$$T^{4} + 2056709392566 T^{6} + T^{8}$$